Presales, Financing Constraints, and
Developers’ Production Decisions
Authors
S u H a n C h a n , Fa n g Fa n g , a n d J i n g Ya n g
Abstract
This study explores the impacts a presale contract has on a
developer’s pricing and production decisions in a gametheoretical framework. In an environment where developers
have full capital market access, the findings reveal that both
developers and buyers are indifferent between a presale and a
spot sale method. However, in an environment with financing
constraints, both developers and buyers are better off when a
presale method is used. This is because the presale method solves
the financing constraint by injecting equity into the development
and, hence, reducing financing costs. This model prediction
seems to describe well the real world situations seen in some of
the property markets in Asia that have nascent financial systems.
The presale method has been a popular tool for selling properties in many Asian
countries for at least the past five decades.1 Under this method, a developer can
sell a property before its completion (or even before its construction). However,
even with more than five decades of experience with this system, we still do not
know much about its benefits (as compared to a spot sale at the completion of a
project), nor its impact on the market structure of the product market. Recently,
the presale practice started to gain increasing attention in the United States as
residential property prices surged during the 2005–2006 period. For example, the
Los Angeles Times (8/22/06) reported that, ‘‘In previous real estate cycles, there’s
been lots of building...and builders and lenders were stuck with a lot of inventory
... But new builders pre-sell just about every building they build. They are much
more cautious.’’2 If so, the news story seems to suggest that the presale system
benefits developers by helping them unload their inventory in a speculative market.
This is probably the case, as it confirms the report by Lai, Wang, and Zhou (2004)
that the presale system offers developers an opportunity to share risks with buyers.
Although there has been growing research on Asian property markets, the presale
method (which is very popular in Asia) has not received much attention in the
academic literature.3 We can find only a few studies that address this issue. One
group of studies treats a presale as a forward or futures contract (ignoring a buyer’s
option to default) and addresses issues such as pricing factors (Chang and Ward,
1993) and the relationship between presale pricing and future spot sale pricing
J R E R
兩
Vo l .
3 0
兩
N o .
3 – 2 0 0 8
3 4 6
兩
C h a n ,
F a n g ,
a n d
Ya n g
(Wong, Yiu, Tse, and Chau, 2006). Another stream of the literature treats a presale
as an option contract, and addresses issues such as its impacts on a developer’s
development strategies (Lai, Wang, and Zhou, 2004) and its influence on market
structure (Wang, Zhou, Chan, and Chau, 2000). Finally, Hua, Chang, and Hsieh
(2001) demonstrate that, in addition to being viewed as a ‘‘stock’’ (inventory of
properties) market, a presale market can be viewed as a ‘‘flow’’ market. The
authors believe that, in the presale market, since trades can be made on properties
in progress, the housing supply can be adjusted to catch the contemporaneous
demand change and economic outlook.
However, many unanswered questions about the presale method still remain. If
the primary function of the presale method is for developers to unload their
inventories in a speculative market (as described in the LA Times article) or for
them to share risks with buyers, then we need to know what impact this practice
might have on other market participants. Will the pre-sold inventories eventually
create problems for those who bought them from the developers? Should buyers
wait and buy the properties in the spot market after the project is completed? Will
developers behave differently in their pricing and production decisions with and
without the presale system? In addition, it is a popular belief in Asia that, since
the debt markets are nascent, the presale system is a necessary tool for developers
to obtain financing. Clearly, none of the existing studies has fully addressed these
issues.
In this paper, we provide a simple equilibrium model in a game-theoretical
framework to throw light on the relationship between presales and developers’
pricing and production decisions.4 In this framework, a presale contract can be
viewed as an option. That is, buyers have the right to buy a property at a
predetermined exercise price and to default on the contract if the spot price dives
below the contract price.5 Under this framework, we find that the presale method
per se does not affect a developer’s production decision. In equilibrium, developers
will increase the presale price to compensate for the option value they give to the
buyers. However, when we model the presale system under the condition that
developers are facing financing constraints, we find a Pareto improvement for
developers and buyers. More importantly, we find that developers will tend to
increase the scale of their developments under the presale system. Our result seems
to conform to the high volume of development activities observed in those Asian
countries that lack a mature financial system, such as China.
We develop our benchmark model in Section 2. In Section 3, we explain why a
presale contract can be viewed as an option, and show the relationship between
the level of downpayment and the presale price. We then model a developer’s
pricing and production decisions under financing constraints in Section 4. Section
5 concludes.
兩
Benchmark Model
In this model, we consider a risk-neutral representative developer who builds and
sells a property within a time period t 僆 [0, 1]. The construction starts at t ⫽ 0
P r e s a l e s ,
F i n a n c i n g
C o n s t r a i n t s ,
a n d
D e v e l o p e r s ’
兩
3 4 7
and finishes at t ⫽ 1. The developer can either pre-sell the property in the presale
market at t ⫽ 0, or wait until t ⫽ 1 to sell the completed property in the spot
market. To make the decision, at t ⫽ 0, the developer decides on 1) the optimal
quantity q and associated price pp that will give him the maximum expected
presale profit E(␲)兩presale, and 2) the optimal quantity qn and associated price pn
that will give him the maximum expected spot sale profit E(␲)兩spot. Comparing
E(␲)兩presale and E(␲)兩spot, the developer chooses the selling approach that brings him
the higher expected profit. The property will then be constructed at a constant
average cost c. We assume that in the presale (but not in the spot sale) the
developer will incur a project-specific fixed cost K, which can be considered as a
special cost of using the presale method.
We also consider a risk-neutral representative buyer for this property. At t ⫽ 0,
after observing the presale price pp, the buyer will decide to either accept or reject
the presale offer. Upon rejection, the buyer needs to wait until t ⫽ 1 to buy the
property in the spot market at price p1. Upon acceptance, however, she needs to
make an immediate downpayment, which is assumed to be a percentage, d, of the
presale price pp. The balance will be paid at t ⫽ 1 upon delivery of the property.
Of course, the buyer can also decide not to purchase the presold property if she
believes that she can buy it cheaper in the open market.
For simplicity, we assume a linear and inverse demand function p1 ⫽ a ⫹ z̃ ⫺
bq, which will be realized at t ⫽ 1. In this function, a is the base demand and z̃
is the market demand shock that remains unknown until t ⫽ 1. However, at t ⫽
0, it is common knowledge that z̃ is in the range [z ⫺ ␴, z ⫹ ␴] with a probability
distribution function ⌽(z̃). b is the property’s price elasticity to quantity. To prevent
a negative market price, we assume that a is significantly larger than other
parameters in the demand function.
Note that at t ⫽ 0, a buyer’s accept-reject decision towards the presale offer is
made under uncertainty about the market demand shock z̃. At t ⫽ 1, when the
market demand shock is realized at z and a market price is formed at p1 ⫽ a ⫹
z ⫺ bq, the buyer’s follow-up decision is either to pay the presale balance, which
equates to the compounded value of the unpaid portion of the presale price,
[(1 ⫺ d)pp](1 ⫹ i), or to default and buy the completed property in the spot market
at price p1. Intuitively, the buyer may have an incentive to default if the realized
demand shock z is low enough to generate an attractive spot price p1. Finally, we
also assume that the risk-free rate i is positive; in other words, there is a time
value of money.
The whole model involves three major steps: (1) the developer’s decision at t ⫽
0 on the optimal development size and price under each selling approach; (2) the
buyer’s decision at t ⫽ 0 on whether or not to accept the presale deal; and (3) the
buyer’s decision at t ⫽ 1 on whether or not to continue with the presale payment
if she has accepted the presale offer. Exhibit 1 summarizes those decision rules
in a step-by-step fashion, while Exhibit 2 summarizes the payoff for each route
taken by the developer and the buyer.
J R E R
兩
Vo l .
3 0
兩
N o .
3 – 2 0 0 8
3 4 8
兩
C h a n ,
F a n g ,
a n d
Ya n g
E x h i b i t 1 兩 Game Tree for the Benchmark Model
Demand shock
~
z ~ Φ[ z − σ , z + σ ]
Developer
Offer presale
t=0
Does not
q and pp
offer presale
Buyer
qn
Accept presale
Reject presale
Downpayment d pp
Demand shock
is realized at z
Demand shock
is realized at z
Demand shock
is realized at z
Buyer
Pay [(1 – d )pp](1 + i)
Default
t=1
Developer
Sell q at
p1 = a + z-bq
PAC
PAS
Developer
Sell q at
p1 = a + z-bq
PR
Sell qn at
pn = a + z-bqn
N
The game covers two time periods: t ⫽ 0 and t ⫽ 1. The demand shock z is unknown until t ⫽ 1. However,
from t ⫽ 0, its distribution z̃ ⬃ ⌽[z ⫺ ␴, z ⫹ ␴] is already known by the public. At t ⫽ 0, a developer decides
to either immediately offer a presale contract with a property size q and a price pp, or build a property with size
qn and put it in the spot market at t ⫽ 1 with market clearing price pn ⫽ a ⫹ z-bqn. Upon a presale, a buyer
decides to either accept the offer and pay the downpayment d pp or reject the offer. At t ⫽ 1, upon accepting
the presale offer, the buyer decides to either pay the compounded value of the balance of the payment [(1 ⫺
d)pp](1 ⫹ i), or default on the payment; upon rejecting the presale offer or accepting it first but defaulting later,
the buyer will switch to the spot market at t ⫽ 1 to buy the property from the developer at market clearing price
p1 ⫽ a ⫹ z-bq. The game ends at terminal nodes PAC, PAS, PR or N, respectively, when a committed presale,
a defaulted presale, a rejected presale or a spot sale occurs.
P r e s a l e s ,
F i n a n c i n g
E x h i b i t 2 兩 Outcomes and Payoffs
PAC
Presale
Accept offer,
commit
Presale
pp
(pp ⫺ c)q ⫺ K
PAS
Presale
Accept offer,
default
Spot sale
PR
Presale
Decline offer
Spot sale
1
(a ⫹ z̃ ⫺ bq)
(1 ⫹ i)
N
Spot sale
Accept offer
Spot sale
1
(a ⫹ z̃ ⫺ bqn)
(1 ⫹ i)
Buyer’s Action
dpp ⫹
1
(a ⫹ z̃ ⫺ bq)
(1 ⫹ i)
冋
冋
冋
dpp ⫹
册
1
(a ⫹ z̃ ⫺ bq) ⫺ c
(1 ⫹ i)
册
册
1
(a ⫹ z̃ ⫺ bq) ⫺ c
(1 ⫹ i)
1
(a ⫹ z̃ ⫺ bqn) ⫺ c
(1 ⫹ i)
q⫺K
q⫺K
qn
兩
N o .
兩
3 4 9
3 – 2 0 0 8
D e v e l o p e r s ’
3 0
Notes: This table summarizes all the possible outcomes and the corresponding payoffs for each buyer and the representative developer. PAC, PAS, PR, and
N are the indexes for outcomes (see Exhibit 1), each matched with a set of decisions; pp and q are the price and quantity under a presale; qn is the quantity
under a spot sale; c is the construction cost per unit; i is the risk-free rate; K is the additional transaction cost associated with presales; z̃ is the uncertain
demand shock; a is the base demand; and b is the price elasticity of quantity.
a n d
Developer’s NPV
兩
Developer’s
Offer Type
C o n s t r a i n t s ,
Each Buyer’s NPV
Vo l .
Final Sale
Type
J R E R
Terminal
Node
3 5 0
兩
C h a n ,
F a n g ,
a n d
Ya n g
The following sub-sections present the major procedure of solving this model with
backward induction. This means we will assume that the buyer has signed a
presale contact and will first look at her decision (at t ⫽ 1) on whether to continue
with the presale contract.
A t t ⴝ 1 : B u y e r ’s D e c i s i o n t o C o n t i n u e o r D e f a u l t o n a
Presale Contract
This subgame exists contingent upon the developer’s presale offer and the buyer’s
presale acceptance. After observing the demand shock z, the buyer will decide to
either pay the presale balance [(1 ⫺ d)pp](1 ⫹ i), or default and buy the property
at its spot price p1. Given that the previous downpayment is a sunk cost, the buyer
will compare the present value of her incremental cost from continuing with the
presale payment versus that from defaulting, or:
⌬C兩continue ⫽ [(1 ⫺ d)pp](1 ⫹ i),
(1)
⌬C兩default ⫽ p1 ⫽ a ⫹ z ⫺ bq.
(2)
Her sufficient and necessary condition to continue with the presale payment is:
⌬C兩continue ⱕ ⌬C兩default, that is:
z ⱖ ẑ ⫽ ⫺ a ⫹ bq ⫹ (1 ⫺ d)(1 ⫹ i)pp,
(3)
where ẑ is the demand shock level that makes the buyer indifferent between these
two options and any demand shock higher than this level will induce a full
payment. Note that we can derive:
⭸ẑ
⫽ ⫺(1 ⫹ i)pp ⬍ 0.
⭸d
(4)
Intuitively, when the downpayment percent d is higher, the loss of downpayment
upon default is also higher. This gives the buyer more incentive to continue with
her payment and to follow through with the presale contract.
P r e s a l e s ,
F i n a n c i n g
C o n s t r a i n t s ,
a n d
D e v e l o p e r s ’
兩
3 5 1
A t t ⴝ 0 : B u y e r ’s D e c i s i o n t o B u y a t P r e s a l e o r S p o t
Market
The existence of this part of the subgame depends on the developer’s decision to
make a presale offer. At t ⫽ 0, the demand shock ẑ is not realized yet. The present
value of the buyer’s expected total cost if she accepts the presale offer is:
冕
⫹冕
E(C)兩accept ⫽
z⫹␴
ẑ
ẑ
z⫺␴
ppd⌽(z̃)
冉
dpp ⫹
冊
1
(a ⫹ z̃ ⫺ bq) d⌽(z̃).
(1 ⫹ i)
(5)
The first integral is the present value of her expected cost if she later (at t ⫽ 1)
chooses to continue with the presale contract. The second integral is the present
value of her expected cost if she later defaults and goes to the spot market. In
contrast, the present value of her expected total cost if she declines the presale
offer at t ⫽ 0 and waits until t ⫽ 1 to buy the property in the spot market is:
E(C)兩decline ⫽
冕
z⫹␴
z⫺␴
1
(1 ⫹ i)
(a ⫹ z̃ ⫺ bq)d⌽(ẑ).
(6)
Her sufficient and necessary condition to accept the presale offer at t ⫽ 0 is:
E(C)兩accept ⱕ E(C)兩decline.
(7)
To simplify the presentation we assume that z̃ follows a uniform distribution on
support [z ⫺ ␴, z ⫹ ␴]. The mathematical result for the buyer’s condition to
accept the presale offer is:
⵩
pp ⱕ pp(q) ⫽
⫹
a ⫹ z ⫺ bq
(1 ⫺ d)(1 ⫹ i)
(1 ⫹ d)␴ ⫺ 2兹d␴[␴ ⫹ (1 ⫺ d)(a ⫹ z ⫺ bq)]
. (8)
(1 ⫺ d)2(1 ⫹ i)
J R E R
兩
Vo l .
3 0
兩
N o .
3 – 2 0 0 8
3 5 2
兩
C h a n ,
F a n g ,
a n d
Ya n g
⵩
That is, if the presale price is no higher than pp(q), then the presale offer will be
accepted.
A t t ⴝ 0 : D e v e l o p e r ’s D e c i s i o n t o U s e a P r e s a l e o r a S p o t
Sale Method
With backward induction, which takes into account the buyer’s decision on
whether to accept the presale contract as well as her decision on whether to
continue with the payment if she has accepted the presale offer, the developer
chooses a selling method (a presale or a spot sale) for his development project.
This choice also includes his decisions on the optimal quantity under each selling
method as well as the price if he were to offer a presale contract.
Consequently, this subgame involves two optimization steps. The first is to find
the optimal quantity (and price) under each of the two selling methods. The second
is to select between the presale method and the spot sale method. With the
previous subgame results given, the developer upon making a presale offer knows
⵩
that the buyer will accept the offer only if the offer package satisfies pp ⱕ pp(q).
This is the case since the developer will not have an incentive to make a presale
offer that is deemed to be rejected. As a result, the only effective options left for
the developer at the beginning of the game is⵩to offer either a spot-sale deal or
an acceptable presale deal that satisfies pp ⱕ pp(q). The developer’s optimization
problem for offering an acceptable presale deal is, therefore:
E(␲)兩presale ⫽ max
{pp,q}
冕
冕
z⫹␴
ẑ
ppqd⌽(z̃)
ẑ
⫹
z⫺␴
[dppq ⫹
1
(1 ⫹ i)
(a ⫹ z̃ ⫺ bq)q]d⌽(z̃)
⫺cq ⫺ K,
(9)
where ẑ ⫽ ⫺a ⫹ bq ⫹ (1 ⫺ d)(1 ⫹ i)pp,
⵩
s.t. pp ⱕ pp(q) ⫽
⫹
a ⫹ z ⫺ bq
(1 ⫺ d)(1 ⫹ i)
(1 ⫹ d)␴ ⫺ 2兹d␴[␴ ⫹ (1 ⫺ d)(a ⫹ z ⫺ bq)]
.
(1 ⫺ d)2(1 ⫹ i)
Here, E(␲)兩presale defines the present value of the developer’s expected payoff from
offering a presale contract, where 兰ẑz⫹␴ ppqd⌽(z̃) is the present value of his
P r e s a l e s ,
F i n a n c i n g
C o n s t r a i n t s ,
a n d
D e v e l o p e r s ’
兩
3 5 3
ẑ
expected development revenue if the buyer continues with the payment, 兰z⫺
␴
[dppq ⫹ 1/(1 ⫹ i) (a ⫹ z̃ ⫺ bq)q]d⌽(z̃) is the present value of the developer’s
expected development revenue if the buyer stops payment at a later stage (under
this circumstance, the developer has to sell the property in the spot market when
it is completed), cq is the development cost for the presale property, and K is the
presale associated cost. Finally, the pricing is subjected to the constraint pp ⱕ
⵩
pp(q).
Alternatively, given the projected demand function (p1 ⫽ a ⫹ z ⫺ bq), the
developer can select an optimal price pn and an optimal quantity qn to sell the
project in the spot market. For this option, the developer will solve:
E(␲)兩spot ⫽ max
{qn}
冕
z⫹␴
z⫺␴
冋冉
1
(1 ⫹ i)
冊 册
pn ⫺ c qn d⌽(z̃),
(10)
where pn ⫽ a ⫹ z̃ ⫺ bqn.
Here, E(␲)兩spot defines the present value of the developer’s expected payoff if he
decides to sell the property on the spot at t ⫽ 1. Note that the buyer’s expected
cost from this spot sale will be:
E(C)兩spot ⫽
冕
z⫹␴
z⫺␴
pn
(1 ⫹ i)
d⌽(z̃),
(11)
where pn ⫽ a ⫹ z̃ ⫺ bqn.
After solving for the optimal prices and quantities under the two selling methods
(presale versus spot sale), the developer compares the expected net present values
for these two choices, E(␲ (p*,
p q*
p ))兩presale versus E(␲ (p*,
n q*
n ))兩spot, and chooses the
presale method only if:
E(␲ (p*,
p q*))
p 兩presale ⱖ E(␲ (p*,
n q*))
n 兩spot.
(12)
Exhibit 3 summarizes the decision rules of the developer and the buyer. The
equilibrium outcomes are presented in Proposition 1.
Proposition 1. In equilibrium, the optimal price and optimal quantity of a presale
contract are:
J R E R
兩
Vo l .
3 0
兩
N o .
3 – 2 0 0 8
3 5 4
兩
C h a n ,
F a n g ,
E x h i b i t 3 兩 Decision Rules
E (␲ (p*,
p q*
p ))兩presale
ⱖ
E (␲ (p*,
n q*
n ))兩spot
Presale
E (␲ (p*,
p q*
p ))兩presale
⬍
E (␲ (p*,
n q*
n ))兩spot
Spot sale
Price Condition
⵩
p*p ⱕ pp(q*)
p
⵩
Buyer’s Response
to Offer
Realized Demand
Shock
Buyer’s Default
Decision
Terminal
Node
Accept
Decline
z ⱖ ẑ
z ⬍ ẑ
—
No default
Default
—
PAC
PAS
PR
Ya n g
Sale Choice
a n d
Developer’s NPV
Comparisons
Accept
—
—
N
p*p ⬎ pp(q*)
p
—
Notes: This table summarizes the decision rules for the developer and the buyer. PAC, PAS, PR, and N are the indexes for outcomes (see Exhibit 1); E (␲) is
the developer’s expected NPV from offering property sales; p*p and q*p are the optimal price and quantity under a presale; q*n is the optimal quantity (with a
⵩
2
corresponding price p*n ) under a spot sale; pp(q*p ) ⫽ a ⫹ z ⫺ bq*p / (1 ⫺ d)(1 ⫹ i) ⫹ (1 ⫹ d)␴ ⫺ 2兹d␴[␴ ⫹ (1 ⫺ d)(a ⫹ z ⫺ bq*)]
p / (1 ⫺ d) (1 ⫹ i) is the
ceiling for the presale price to avoid a rejection from the buyer; and ẑ ⫽ ⫺a ⫹ bq*p ⫹ (1 ⫺ d)(1 ⫹ i)p*p is the ceiling for the demand shock to avoid a
default on presale payment.
P r e s a l e s ,
F i n a n c i n g
C o n s t r a i n t s ,
q*p ⫽
a ⫹ z ⫺ c(1 ⫹ i)
, and
2b
p*p ⫽
a ⫹ z ⫹ c(1 ⫹ i)
2(1 ⫺ d)(1 ⫹ i)
⫹
a n d
D e v e l o p e r s ’
兩
3 5 5
(13)
␴ (1 ⫹ d) ⫺ 兹2d␴[(1 ⫺ d)(a ⫹ z ⫹ c(1 ⫹ i)) ⫹ 2␴]
.
(1 ⫺ d)2(1 ⫹ i)
(14)
The optimal price and optimal quantity of a spot sale contract are:
q*n ⫽
a ⫹ z ⫺ c(1 ⫹ i)
, and
2b
(15)
p*n ⫽
a ⫹ z ⫹ c(1 ⫹ i)
.
2
(16)
Thus, the development scale is the same under both selling methods, with:
q*p ⫽ q*.
n
(17)
Ignoring the presale-specific transaction cost K, a developer is indifferent between
offering a presale or a spot sale, with:
E(␲ (p*,
p q*))
p 兩presale ⫽ E(␲ (p*,
n q*))
n 兩spot
⫽
[a ⫹ z ⫺ c(1 ⫹ i)]2
.
4b(1 ⫹ i)
(18)
Finally, a buyer’s expected cost of buying the property should be the same
regardless of the selling method used by the developer, with:
E(C(p*,
p q*))
p 兩accept ⫽ E(C(p*,
p q*))
p 兩decline
⫽ E(C(p*,
n q*))
n 兩spot ⫽
J R E R
a ⫹ z ⫹ c(1 ⫹ i)
.
2(1 ⫹ i)
兩
Vo l .
3 0
兩
N o .
(19)
3 – 2 0 0 8
3 5 6
兩
C h a n ,
F a n g ,
a n d
Ya n g
Proof. See Appendix A. 䡵
These results are very interesting. When a presale serves only as a method to sell
properties, it does not affect the fundamental supply and demand in the underlying
market (meaning that developers will not alter their production decisions based
on the presale result, nor will buyers change their demand curves). Under this
scenario, the presale method will not affect the development scale (and the
associated marketclearing price), the market structure, nor the developer’s welfare.
When the presale-specific transaction cost is very trivial, the developer will
be indifferent between a presale contract and a spot sale. In other words, when
the presale-specific transaction cost is non trivial, the developer will prefer a spot
sale.
In addition, a presale will not impact the buyer’s welfare either. As compared to
a spot sale, a presale will provide the buyer flexibility after a demand shock occurs.
If the demand shock is small such that the spot price is relatively lower than the
presale commitment (i.e., the compounded residual payments for the presale), the
buyer will default on the presale payment and switch to the spot market. However,
this flexibility is not free and its costs include a downpayment and a price
difference between the two contracts, which we will discuss in the next section.
In an efficient market, this flexibility will be fully priced, and as such the buyer’s
expected purchase cost will be the same as in a spot sale.
兩
Downpayment as an Option
In essence, a presale contract gives a buyer a call option (at t ⫽ 0) to buy the
completed project in the next period (at t ⫽ 1) at a contract price (which should
be lower than the spot market price). Given this, the downpayment can be
considered as a part of the cost of this option, while the strike price is the residual
payment specified in the presale contract. When the project is completed at t ⫽
1, a spot price higher than the residual payment will trigger an exercise of this
call option. In contrast, if the spot price is lower than the residual payment, then
the call option will not be exercised. Under this circumstance, the buyer will buy
the property in the spot market. Given this, it is interesting to investigate the
relationships between the presale downpayment and the presale-spot sale price
spread.
Option and its Cost
Comparing the presale price [in Equation (14)] and the spot price [in Equation
(16)], we derive the present value of the presale-spot sale price spread as:
P r e s a l e s ,
F i n a n c i n g
⌬p ⫽ p*
p ⫺
⫽
1
(1 ⫹ i)
C o n s t r a i n t s ,
a n d
D e v e l o p e r s ’
兩
3 5 7
p*n
1
{(1 ⫺ d)d[a ⫹ z ⫹ c(1 ⫹ i)]
2(1 ⫺ d)2(1 ⫹ i)2
⫹ 2(1 ⫹ i)␴ (1 ⫹ d)
⫺ 2(1 ⫹ i)兹2d␴[(1 ⫺ d)(a ⫹ z ⫹ c(1 ⫹ i)) ⫹ 2␴]} ⱖ 0.
(20)
This is true because:
[(1 ⫺ d)d[a ⫹ z ⫹ c(1 ⫹ i)] ⫹ 2(1 ⫹ i)␴ (1 ⫹ d)]2
⫺ [2(1 ⫹ i)兹2d␴[(1 ⫺ d)(a ⫹ z ⫹ c(1 ⫹ i)) ⫹ 2␴]]2
⫽ (1 ⫺ d)2(1 ⫹ i)2[d(a ⫹ z ⫹ c(1 ⫹ i)) ⫺ 2␴]2 ⱖ 0.
(21)
Equation (20) suggests that, when the percentage of downpayment d is lower than
100%, ceteris paribus, a presale contract is more expensive than a projected spot
sale contract (in present value terms). However, when d is 100%, the presale price
is identical to the present value of the projected spot price. In this scenario, the
option value is zero since the strike price is also zero. Given this, the higher the
presale downpayment, the lower the value of the option and the closer the prices
between the presale contract and the spot sale contract.
To see this, from Equation (19), we derive:
ƒ ⫽ E(C)(p*,
p q*))
p 兩accept ⫺
a ⫹ z ⫹ c(1 ⫹ i)
⫽ 0.
2(1 ⫹ i)
(22)
Based on Roy’s Identity:
⭸E(C(p*,
⭸ƒ
p q*))
p 兩accept
dp*p
⭸d
⭸d
⫽⫺
⫽⫺
⬍ 0.
⭸E(C( p*,
dd
⭸ƒ
p q*))
p 兩accept
⭸p*
⭸p*
p
p
J R E R
兩
Vo l .
(23)
3 0
兩
N o .
3 – 2 0 0 8
3 5 8
兩
C h a n ,
F a n g ,
a n d
Ya n g
Intuitively, a buyer’s cost in a presale contract increases in the presale price and
the downpayment percentage. That is, ⭸E(C(p*,
p q*
p ))兩accept / ⭸d ⬎ 0 and ⭸E(C(p*,
p
q*
p ))兩accept / ⭸p*
p ⬎ 0. Given this, Equation (23) results in dp*
p /dd ⬍ 0. Since spot
price p*n is unaffected by presale downpayment d, we know that:
d⌬p
⫽
dd
冉
d p*p ⫺
1
(1 ⫹ i)
dd
冊
p*
n
⫽
dp*
p
⬍ 0.
dd
(24)
The negative interaction between the downpayment and the price spread (p*p ⫺ 1/
(1 ⫹ i) p*
n ) indicates a substitution relationship. In other words, to attract a buyer
to accept a presale contract with a higher downpayment, the developer will have
to reduce the presale price to compensate for the loss of some option value.
Because of this substitution effect, the downpayment ratio will only affect the
price of the presale contract, but not the developer’s profit (and, hence, his
production decision). This relationship can be easily seen from Equations (18),
(19), and (17), where d does not have an impact on E(␲ (p*,
p q*
p ))兩presale, E(␲ (p*,
n
q*
n ))兩spot, E(C)(p*,
p q*
p ))兩accept, E(C(p*,
p q*
p ))兩decline, q*
p , or q*.
n
Refund Policy
In the real world, it is not uncommon to find a buyer in a presale contract asking
the developer to refund all or part of the downpayment when the property price
drops below the presale price at the time of exercise. The buyer normally does
this by complaining about construction quality and/or by accusing the developer
of not building the property according to the specifications listed in the contract.
It is possible that a developer is willing to refund part of the downpayment just
to avoid a long legal battle and/or possible reputation damage.
Given this, we now analyze the developer’s decision under a refund possibility.
We assume that the refund policy is known to both the buyer and the developer
at the beginning period (t ⫽ 0) and the refund is calculated as a percentage x of
the initial downpayment.
With a refund percentage x, the buyer’s decision at t ⫽ 1 on whether to default
involves comparing the present values of her incremental costs associated with
the two choices:
P r e s a l e s ,
F i n a n c i n g
C o n s t r a i n t s ,
a n d
D e v e l o p e r s ’
⌬C兩continue ⫽ [(1 ⫺ d)pp](1 ⫹ i), and
兩
3 5 9
(25)
⌬C兩default ⫽ p1 ⫽ a ⫹ z ⫺ bq ⫺ xdpp.
(26)
The necessary and sufficient condition to continue with the presale payment is
⌬C兩continue ⱕ ⌬C兩default, or:
z ⱖ ẑ ⫽ ⫺a ⫹ bq ⫹ [(1 ⫺ d⬘)(1 ⫹ i)]pp, where
冉
d⬘ ⫽ d 1 ⫺
(27)
冊
x
.
1⫹i
(28)
Comparing the demand shock turning point ẑ defined in Equation (27) with that
defined in Equation (3), we can see that the only difference is that d is replaced
by d⬘.
With a refund, the present value of a buyer’s expected total cost if she accepts
the presale offer is:
E(C)兩accept ⫽
冕
z⫹␴
ẑ
ppd⌽(z̃) ⫹
冉
dpp ⫹
⫽
冕
z⫹␴
ẑ
冉
1
(1 ⫹ i)
ppd⌽(z̃) ⫹
d⬘pp ⫹
冕
ẑ
z⫺␴
冊
(a ⫹ z̃ ⫺ bq ⫺ xdpp) d⌽(z̃)
冕
ẑ
z⫺␴
冊
1
(a ⫹ z̃ ⫺ bq) d⌽(z̃),
(1 ⫹ i)
(29)
which differs from the expected cost without refund [as shown in Equation (5)]
only in that d is replaced by d⬘. Consequently, the decision rule with a refund
should also be similar to that with no refund [as shown in Equation (8)].
J R E R
兩
Vo l .
3 0
兩
N o .
3 – 2 0 0 8
3 6 0
兩
C h a n ,
F a n g ,
a n d
Ya n g
With a refund, the present value of the developer’s presale offer is:
E(␲)兩presale ⫽ max
{pp,q}
冕
冕
z⫹␴
ẑ
ppqd⌽(z̃)
ẑ
⫹
z⫺␴
1
(a ⫹ z̃ ⫺ bq ⫺ xdpp)q]d⌽(z̃)
(1 ⫹ i)
[dppq ⫹
⫺ cq ⫺ K
⫽ max
{pp,q}
冕
冕
z⫹␴
ẑ
ppqd⌽(z̃)
ẑ
⫹
z⫺␴
[d⬘ppq ⫹
1
(1 ⫹ i)
(a ⫹ z̃ ⫺ bq)q]d⌽(z̃)
⫺ cq ⫺ K,
(30)
where
ẑ ⫽ ⫺a ⫹ bq ⫹ (1 ⫺ d⬘)(1 ⫹ i)pp,
冉
d⬘ ⫽ d 1 ⫺
⵩
s.t. pp ⱕ pp(q) ⫽
⫹
冊
x
,
1⫹i
a ⫹ z ⫺ bq
(1 ⫺ d⬘)(1 ⫹ i)
(1 ⫹ d⬘)␴ ⫺ 2兹d⬘␴[␴ ⫹ (1 ⫺ d⬘)(a ⫹ z ⫺ bq)]
.
(1 ⫺ d⬘)2(1 ⫹ i)
This differs from the present value of the developer’s presale offer (without a
refund) only in that d is replaced by d⬘. In fact, a refund actually reduces the
downpayment ratio from d to d⬘ ⫽ d(1 ⫺ x/(1 ⫹ i). Given this, the equilibrium
presale price in Equation (14) is changed to:
p*p ⫽
⫹
a ⫹ z ⫹ c(1 ⫹ i)
2(1 ⫺ d⬘)(1 ⫹ i)
␴ (1 ⫹ d⬘) ⫺
兹2d⬘␴[(1 ⫺ d⬘)(a ⫹ z ⫹ c(1 ⫹ i)) ⫹ 2␴]
冉
where d⬘ ⫽ d 1 ⫺
(1 ⫺ d⬘)2(1 ⫹ i)
x
冊
1⫹i
.
,
(31)
P r e s a l e s ,
F i n a n c i n g
C o n s t r a i n t s ,
a n d
D e v e l o p e r s ’
兩
3 6 1
Following Equations (23) and (24), we find that:
dp*p
⬍ 0,
dd⬘
d⌬p
⫽
dd⬘
(32)
冉
d p*p ⫺
冊
1
p*
(1 ⫹ i) n
dd⬘
⫽
dp*
p
⬍ 0,
dd⬘
(33)
and correspondingly, the impacts of a refund percentage x on the presale price
and its spread from the spot price are:
dp*p
dp*
dd⬘
p
⫽
䡠
⬎ 0,
dx
dd⬘ dx
d⌬p
⫽
dx
冉
d p*p ⫺
(34)
冊
1
p*
(1 ⫹ i) n
dx
⫽
dp*
p
⬎ 0, given
dx
(35)
dd⬘
d
⫽⫺
⬍ 0.
dx
1⫹i
(36)
Equation (34) indicates that, if there is a refund, the presale price and its spread
from the spot price are positively affected by the refund percentage x. That is,
dp*/dx ⬎ 0 and d⌬p/dx ⬎ 0. This is true because a refund from a developer
(when a buyer defaults) is equivalent to a reduction in the downpayment of the
presale contract. If a developer refunds part of the downpayment, this refund
actually reduces the buyer’s default cost and the developer’s benefit from receiving
the ‘‘lock-in’’ downpayment. Of course, this is equivalent to a reduction in the
original downpayment ratio. Indeed, if a developer allows a buyer to default with
a lower cost, the developer will have to increase the presale price in order to
compensate for his expected risk increase. This means that the increased risk
(associated with a refund) assumed by the developer is fully priced into the presale
contract.
Extreme Cases
In this sub-section, we discuss two special cases. The first is when the
downpayment ratio is zero and the second is when the ratio is positive but buyers
will receive a full refund upon default (i.e., both the downpayment and its interest
will be returned to the buyer at t ⫽ 1). These two cases are interesting in that
J R E R
兩
Vo l .
3 0
兩
N o .
3 – 2 0 0 8
3 6 2
兩
C h a n ,
F a n g ,
a n d
Ya n g
they both will reduce the effective downpayment to a minimum. We will see how
a presale option will be priced under these circumstances.
In the benchmark model, when the downpayment ratio d ⫽ 0, Equation (14) is
reduced to:
p*p 兩zero-down ⫽
a ⫹ z ⫹ c(1 ⫹ i) ⫹ 2␴
2(1 ⫹ i)
(37)
⫽ max p*(d).
p
(38)
{d}
Correspondingly,
⌬p兩zero-down ⫽ p*
p 兩zero-down ⫺
1
(1 ⫹ i)
p*
n ⫽
␴
1⫹i
⫽ max ⌬p(d).
(39)
(40)
{d}
Equation (38) is true because dp*p /dd ⬍ 0 [see Equation (23)]. Consequently, p*p
is highest when d ⫽ 0. As expected, when developers do not receive any up-front
‘‘lock-in’’ downpayment, their default cost reaches a maximum. Since the default
cost will be priced to the maximum, the presale price must be set at the maximum
level of the estimated price range. This conforms to the substitution relationship
between the presale price and the level of the downpayment.
In the model with a downpayment refund, the total refund at t ⫽ 1 (when there
is a full refund upon default) will be xdpp ⫽ (1 ⫹ i)dpp, with the effective refund
ratio being x ⫽ (1 ⫹ i). Intuitively, this will lead to a zero effective downpayment,
which generates an equilibrium presale price similar to that in Equation (37). We
can confirm this result by substituting x ⫽ (1 ⫹ i) into Equation (31):
p*p 兩full-refund ⫽
a ⫹ z ⫹ c(1 ⫹ i) ⫹ 2␴
2(1 ⫹ i)
⫽ max p*(x),
when x ⫽ max{x} ⫽ (1 ⫹ i).
p
{x}
(41)
(42)
P r e s a l e s ,
F i n a n c i n g
C o n s t r a i n t s ,
a n d
D e v e l o p e r s ’
兩
3 6 3
Correspondingly,
⌬p兩full-refund ⫽ p*
p 兩full-refund ⫺
1
(1 ⫹ i)
p*n ⫽
␴
1⫹i
(43)
⫽ max ⌬p(x).
(44)
{x}
Equation (42) holds because dp*
p /dx ⬎ 0 [see Equation (34)]. Consequently, p*
p
is highest when x ⫽ max{x} ⫽ (1 ⫹ i). These two special cases further
demonstrate a commonly known principle that there is no free lunch. A less
stringent downpayment (with refund) policy must end up with a less favorable
contract price. A more favorable contract price must be accompanied by a tougher
downpayment policy.6
At this moment, our model implicitly assumes that the presale method does
not affect a developer’s production decision (and hence, the market structure).
However, it is possible that, when there is a financing constraint on a developer’s
equity side, the developer’s production decision will be affected. This is the case
because there is a cost saving if the developer can receive a downpayment before
the project starts. This issue is examined in the next section.
兩
Financing Constraints
We now consider that the function of a presale is to help developers raise equity
for their developments. In Asia, where debt financing is hard to get, it is common
knowledge that the motivation for developers to employ the presale method is to
raise equity capital. In the benchmark model, we ignore this possibility and assume
that a developer can finance the development with his own equity or with debt if
it can be obtained at a reasonable cost.
In the real world, developers have to raise debt to finance their developments. It
should be noted that, in many Asian countries, the financing cost could be high
because of the tight capital markets. More importantly, debt financing might not
be available at all to developers who do not have adequate equity capital. In
addition, the cost of debt, if debt is available, is a function of the amount of equity
in the project. With a presale, a developer can receive the downpayments of presale
contracts before project construction starts. This cash infusion will help increase
the developer’s equity contribution and, hence, reduce his debt financing cost. On
the other hand, if we keep the leverage ratio constant and use the downpayment
as equity, a developer can increase the size of his development project.
J R E R
兩
Vo l .
3 0
兩
N o .
3 – 2 0 0 8
3 6 4
兩
C h a n ,
F a n g ,
a n d
Ya n g
We now revise our benchmark model to take into account financing constraints.
We define the total development cost as cq. If we assume that the developer’s
only equity source to finance the development is the buyer’s downpayment, then
cdq is financed with equity. The rest of the development cost will be financed
with debt at an interest rate r. The cost of debt is c(1 ⫺ d)(1 ⫹ r)q. We assume
r ⫽ r0 ⫺ wd, where r0 is the baseline interest rate and w is the sensitivity of
interest rate reduction to the presale downpayment ratio (or the leverage ratio).
This means that, the higher the equity level, the lower the interest rate will be for
the development project. The developer’s development cost can be rewritten as:
cdq ⫹ c(1 ⫺ d)(1 ⫹ r)q ⫽ c[d ⫹ (1 ⫺ d)(1 ⫹ r0 ⫺ wd)]q. (45)
Note that if we set w ⫽ 0 and r0 ⫽ 0 (i.e., debt is interest free), the cost function
will collapse to that used in the benchmark model.
It is clear that the development-cost reduction affects a developer’s production and
pricing decisions, but not the buyer’s decision. With backward induction, the
buyer’s decision rules are the same as before. At t ⫽ 1, she will choose to continue
with the presale payment if:
z ⱖ ẑ ⫽ ⫺a ⫹ bq ⫹ (1 ⫺ d)(1 ⫹ i)pp;
(46)
At t ⫽ 0, she will accept the presale offer if:
⵩
pp ⱕ pp(q) ⫽
⫹
a ⫹ z ⫺ bq
(1 ⫺ d)(1 ⫹ i)
(1 ⫹ d)␴ ⫺ 2兹d␴[␴ ⫹ (1 ⫺ d)(a ⫹ z ⫺ bq)]
. (47)
(1 ⫺ d)2(1 ⫹ i)
However, the developer’s optimization problem for offering an acceptable presales
deal is now changed to:
P r e s a l e s ,
F i n a n c i n g
C o n s t r a i n t s ,
冕
z⫹␴
E(␲)兩presale ⫽ max
冕
ẑ
⫹
[dppq ⫹
z⫺␴
D e v e l o p e r s ’
兩
3 6 5
ppqd⌽(z̃)
ẑ
{pp,q}
a n d
1
(a ⫹ z̃ ⫺ bq)q]d⌽(z̃)
(1 ⫹ i)
(48)
⫺c[d ⫹ (1 ⫺ d)(1 ⫹ r0 ⫺ wd)]q ⫺ K,
where
ẑ ⫽ ⫺a ⫹ bq ⫹ (1 ⫺ d)(1 ⫹ i)pp,
⵩
s.t. pp ⱕ pp(q) ⫽
⫹
a ⫹ z ⫺ bq
(1 ⫺ d)(1 ⫹ i)
(1 ⫹ d)␴ ⫺ 2兹d␴[␴ ⫹ (1 ⫺ d)(a ⫹ z ⫺ bq)]
.
(1 ⫺ d)2(1 ⫹ i)
Note that the marginal development cost c in the benchmark model changes to c
[d ⫹ (1 ⫺ d)(1 ⫹ r0 ⫺ wd)]. The developer’s optimization problem under a spot
sale is similar to the process we used in the benchmark model, except that now
we also consider financing constraints, or:
E(␲)兩spot ⫽ max
{qn}
冕
z⫹␴
z⫺␴
冋
1
(1 ⫹ i)
册
pnqn ⫺ c(1 ⫹ r0)qn d⌽(z̃),
(49)
where pn ⫽ a ⫹ z̃ ⫺ bqn.
The equilibrium outcomes of the case are presented in Proposition 2.
Proposition 2. In equilibrium, the optimal price and optimal quantity of a presale
contract (with financing constraints) are:
q*p ⫽
a ⫹ z ⫺ c␺ (1 ⫹ i)
,
2b
p*p ⫽
a ⫹ z ⫹ c␺ (1 ⫹ i)
2(1 ⫺ d)(1 ⫹ i)
⫹
(50)
␴ (1 ⫹ d) ⫺
兹2d␴[(1 ⫺ d)(a ⫹ z ⫹ c␺ (1 ⫹ i)) ⫹ 2␴]
,
(1 ⫺ d)2(1 ⫹ i)
(51)
where ␺ ⫽ d ⫹ (1 ⫺ d)(1 ⫹ r0 ⫺ wd).
J R E R
兩
Vo l .
3 0
兩
N o .
3 – 2 0 0 8
3 6 6
兩
C h a n ,
F a n g ,
a n d
Ya n g
The optimal price and optimal quantity of a spot sale contract are:
q*n ⫽
a ⫹ z ⫺ c(1 ⫹ r0)(1 ⫹ i)
,
2b
(52)
p*n ⫽
a ⫹ z ⫹ c(1 ⫹ r0)(1 ⫹ i)
.
2
(53)
The development scale is greater under a presale contract, with:
q*p ⬎ q*.
n
(54)
Ignoring the presale-specific transaction cost K, a developer is better off offering
a presale contract than a spot sale contract, with:
E(␲ (p*,
p q*))
p 兩presale ⬎ E(␲ (p*,
n q*))
n 兩spot.
(55)
A buyer’s expected cost of buying the property is also lower under a presale
contract than a spot sale, with:
E(C(p*,
(56)
p q*))
p 兩accept ⫽ E(C(p*,
p q*))
p 兩decline ⬍ E(C(p*,
n q*))
n 兩spot.
Finally, the development scale and a developer’s profit are positively affected by
the sensitivity of the debt cost to the leverage ratio (w), while a buyer’s expected
cost is negatively related to w, or:
d⌬q
d⌬E(␲)
d⌬E(C)
⬎ 0,
⬎ 0 and
⬍ 0.
dw
dw
dw
(57)
Proof. See Appendix B. 䡵
Proposition 2 highlights the importance of the presale method in a market with
financing constraints. With a presale method, a developer is able to obtain cash
before a project actually starts. The cash inflows associated with the downpayment
of a presale contract can be used by developers as the equity for the development.
Since developers have more equity with a presale method, it is easier for them
P r e s a l e s ,
F i n a n c i n g
C o n s t r a i n t s ,
a n d
D e v e l o p e r s ’
兩
3 6 7
to obtain debt financing at a lower interest rate. This reduces the marginal
development cost of the project. Under this circumstance, the developer enjoys
additional cost savings when compared with the use of the spot sale method. When
the developer shares the savings with the buyer, the buyer can buy the property
at a price lower than if there were no presale contract. Therefore, both the
developer and the buyer are better off with the presale method. This is probably
why the presale method is more popular in those Asian countries with less
developed financial systems.
It is important to note that this cost-saving efficiency can also result in a larger
development scale. Holding the leverage ratio of a development constant, the cash
inflow obtained from the presale method allows a developer to increase the
development size. This means that we will observe more developments in a market
in which the presale method is used than in one that only allows spot sales. Given
this, in property markets with tight capital constraints (such as in China) where
the underlying motivation for employing the presale method is to overcome
financing constraints, we should observe aggressive development patterns. This
seems to be the phenomenon we have been observing in these markets in recent
years.
兩
Conclusion
The presale method has frequently been used in property markets all over the
world, particularly in Asia. However, even with more than five decades of
experience with this system, we still do not know much about its impact on
property markets, except that developers can use the method to share the
development risk with buyers (see Lai, Wang, and Zhou, 2004).
In this paper, we develop a model to explore the impacts of the presale method
on a developer’s pricing and production decisions. We find that the presale method
itself does not affect a developer’s production decision, nor the welfare of a buyer
when there are no financing constraints in the market. In an efficient market,
developers will simply adjust the presale price to account for the option value
they give to buyers in the presale market and, as such, both developers and buyers
are indifferent between a presale method and a spot sale method.
However, in a market with a nascent financial system where debt financing may
not be available to developers at a reasonable cost, the presale system is beneficial
to both developers and buyers. This is the case because the downpayment from a
presale contract can be used as equity for a development. Under this scenario, we
show that developers will pursue aggressive development strategies. This result
seems to be consistent with the aggressive building behaviors of developers in
some of the Asian real estate markets.
The popular press also mentions that the most important reason for the existence
of the presale system is buyers’ belief that if they do not buy now, they will not
be able to buy in the future. In other words, the ‘‘last bus’’ fear drives buyers to
J R E R
兩
Vo l .
3 0
兩
N o .
3 – 2 0 0 8
3 6 8
兩
C h a n ,
F a n g ,
a n d
Ya n g
the presale market. While this argument makes some sense, it is difficult to model
it under a rational framework.7 However, it may be worthwhile for future research
on the presale system to address this kind of behavioral argument.
兩
兩兩
Appendix
A. Proof for Proposition 1
Our first attempt is to simultaneously solve for the optimal p*
p and q. The
decomposition of Equation (9) with K ⫽ 0 generates:
E(␲)兩presale ⫽
1
4(1 ⫹ i)z
(A1pp2 ⫹ A2pp ⫹ B1q3 ⫹ B2q2 ⫹ B3q ⫹ C),
(58)
where A1 ⫽ ⫺(1 ⫺ d)2(1 ⫹ i)2q ⬍ 0,
B ⫽ ⫺b2 ⬍ 0, and A2, B2, B3 and C
are the combinations of parameters.
The unconstrained optimum p0 is derived from a first-order condition:
dE(␲)兩presale
o
⫽ 0 ⇒ p*
p ⫽ p0
dp0
⫽
(1 ⫺ d)(a ⫹ z ⫺ bq) ⫹ (1 ⫹ d)␴
,
(1 ⫹ i)(1 ⫺ d)2
(59)
and this result must maximize the expected profit since the second-order condition
satisfies:
d2E(␲)兩presale
(1 ⫺ d)2(1 ⫹ i)q
⫽
⫺
⬍ 0.
dp20
2z
(60)
P r e s a l e s ,
F i n a n c i n g
C o n s t r a i n t s ,
a n d
D e v e l o p e r s ’
兩
3 6 9
⵩
However, comparing po0 with pp(q) defined in Equation (8), we find:
⵩
po0 ⫺ pp(q) ⫽
2兹d␴[␴ ⫹ (1 ⫺ d)(a ⫹ z ⫺ bq)]
⬎ 0.
(1 ⫺ d)2(1 ⫹ i)
(61)
Since
the unconstrained optimum po0 exceeds the upper boundary constraint
⵩
pp(q), this unconstrained optimum cannot be a feasible solution for the problem.
Given the observation that E(␲)兩presale is a concave function of pp, we know that
the optimum p*p should be the constrained maximum level of pp. Consequently:
⵩
p*p ⫽ pp(q) ⫽
⫹
a ⫹ z ⫺ bq
(1 ⫺ d)(1 ⫹ i)
(1 ⫹ d)␴ ⫺ 2兹d␴[␴ ⫹ (1 ⫺ d)(a ⫹ z ⫺ bq)]
.
(1 ⫺ d)2(1 ⫹ i)
(62)
The optimum q, however, cannot be solved since E(␲)兩presale is a cube function of
q and the multiplier for term q3 is negative. (Under this circumstance, we can only
find a local minimum, but not any local or global maximum.)
We hence try to solve for the sequential optima starting from q to p*.
With
p
backward induction, we derive the optimum p*p first. Following steps similar to
those mentioned above, we derive the optimum p*
p as shown in Equation (62).
Then we solve for the optimal q by incorporating optimal p*p into the expected
payoff function, resulting in:
E(␲ (p*(q),
q))兩presale ⫽ q
p
冉
冊
a ⫹ z ⫺ bq
⫺ c ⫺ K.
1⫹i
(63)
The optimum q is then derived from a first-order condition:
dE(␲ (p*(q),
q))兩presale
a ⫹ z ⫺ c(1 ⫹ i)
p
⫽ 0 ⇒ q ⫽ q* ⫽
,
dq
2b
(64)
and this result must maximize the expected profit since the second-order condition
satisfies:
J R E R
兩
Vo l .
3 0
兩
N o .
3 – 2 0 0 8
3 7 0
兩
C h a n ,
F a n g ,
a n d
Ya n g
d2E(␲ (p*(q),
q))兩presale
2b
p
⫽⫺
⬍ 0.
2
dq
(1 ⫹ i)
(65)
Substituting q* into Equation (62), we derive the optimal presale price as:
p*p ⫽
⫹
a ⫹ z ⫹ c(1 ⫹ i)
2(1 ⫺ d)(1 ⫹ i)
␴ (1 ⫹ d) ⫺ 兹2d␴[(1 ⫺ d)(a ⫹ z ⫹ c(1 ⫹ i)) ⫹ 2␴]
. (66)
(1 ⫺ d)2(1 ⫹ i)
From the optimization problem under the spot sale [in Equation (10)], it is easy
to derive that the optimal spot sale price and quantity are:
q*n ⫽
a ⫹ z ⫺ c(1 ⫹ i)
, and
2b
(67)
p*n ⫽
a ⫹ z ⫹ c(1 ⫹ i)
.
2
(68)
Finally, we substitute q* and p*
p into Equations (5), (6), and (9), and q*
n and p*
n
into Equations (10) and (11). We find that when the presale-specific transaction
cost K ⫽ 0:
E(␲ (p*,
p q*))
p 兩presale ⫽ E(␲ (p*,
n q*))
n 兩spot
⫽
[a ⫹ z ⫺ c(1 ⫹ i)]2
.
4b(1 ⫹ i)
(69)
That is, the developer is indifferent between offering a presale and offering a spot
sale. In addition, since:
P r e s a l e s ,
F i n a n c i n g
C o n s t r a i n t s ,
a n d
D e v e l o p e r s ’
兩
3 7 1
E(C(p*,
p q*))
p 兩accept ⫽ E(C(p*,
p q*))
p 兩decline
⫽ E(C(p*,
n q*))
n 兩spot ⫽
a ⫹ z ⫹ c(1 ⫹ i)
,
2(1 ⫹ i)
(70)
it suggests that a buyer’s expected cost of buying the property is the same
regardless of the selling method used by the developer. 䡵 Q.E.D.
兩
B. Proof for Proposition 2
To solve this problem, we use a procedure similar to the one used to solve for
the equilibrium solution for the benchmark model. With financing constraints and
a possible development cost reduction, we solve the optimization problem as
detailed in Equation (48). Clearly, the equilibrium presale quantity and price are:
q*p ⫽
a ⫹ z ⫺ c␺ (1 ⫹ i)
,
2b
p*p ⫽
a ⫹ z ⫹ c␺ (1 ⫹ i)
2(1 ⫺ d)(1 ⫹ i)
⫹
(71)
␴ (1 ⫹ d)
⫺兹2d␴[(1 ⫺ d)(a ⫹ z ⫹ c␺ (1 ⫹ i)) ⫹ 2␴]
,
(1 ⫺ d)2(1 ⫹ i)
(72)
where ␺ ⫽ d ⫹ (1 ⫺ d)(1 ⫹ r0 ⫺ wd).
The developer’s expected profits and the buyer’s expected cost under a presale
contract are now changed to:
E(␲ (p*,
p q*))
p 兩presale ⫽
[a ⫹ z ⫺ c␺ (1 ⫹ i)]2
⫺ K,
4b(1 ⫹ i)
(73)
E(C(p*,
p q*))
p 兩accept ⫽ E(C(p*,
p q*))
p 兩decline
⫽
a ⫹ z ⫹ c␺ (1 ⫹ i)
.
2(1 ⫹ i)
(74)
By solving the optimization problem in Equation (49), we can derive that, with
financing constraints, the equilibrium spot sale quantity and spot sale price are:
J R E R
兩
Vo l .
3 0
兩
N o .
3 – 2 0 0 8
3 7 2
兩
C h a n ,
F a n g ,
a n d
Ya n g
q*n ⫽
a ⫹ z ⫺ c(1 ⫹ r0)(1 ⫹ i)
, and as a result,
2b
(75)
p*n ⫽
a ⫹ z ⫹ c(1 ⫹ r0)(1 ⫹ i)
.
2
(76)
The corresponding developer’s expected profits and the buyer’s expected costs are:
E(␲ (p*,
n q*))
n 兩spot ⫽
[a ⫹ z ⫺ c(1 ⫹ r0)(1 ⫹ i)]2
,
4b(1 ⫹ i)
(77)
E(C(p*,
n q*))
n 兩spot ⫽
a ⫹ z ⫹ c(1 ⫹ r0)(1 ⫹ i)
.
2(1 ⫹ i)
(78)
Comparing the production quantities under the two selling methods, we derive:
⌬q ⫽ q*
p ⫺ q*
n
⫽
a ⫹ z ⫺ c␺ (1 ⫹ i) a ⫹ z ⫺ c(1 ⫹ r0)(1 ⫹ i)
⫺
2b
2b
⫽
c(1 ⫹ r0 ⫺ ␺)(1 ⫹ i)
⬎ 0.
2b
(79)
This is true because:
1 ⫹ r0 ⫺ ␺ ⫽ 1 ⫹ r0 ⫺ [d ⫹ (1 ⫺ d)(1 ⫹ r0 ⫺ wd)]
⫽ d[r0 ⫹ w(1 ⫺ d)] ⬎ 0.
(80)
Equation (79) indicates that a presale contract is associated with a larger
production scale than a spot sale contract.
To derive the decision rules for a presale with financing constraints, when K ⫽
0, we obtain:
P r e s a l e s ,
F i n a n c i n g
C o n s t r a i n t s ,
a n d
D e v e l o p e r s ’
兩
3 7 3
⌬E(␲) ⫽ E(␲ (p*,
p q*))
p 兩presale ⫺ E(␲ (p*,
n q*))
n 兩spot
⫽
[a ⫹ z ⫺ c␺ (1 ⫹ i)]2 [a ⫹ z ⫺ c(1 ⫹ r0)(1 ⫹ i)]2
⫺
4b(1 ⫹ i)
4b(1 ⫹ i)
⫽
[2(a ⫹ z) ⫺ c(␺ ⫹ 1 ⫹ r0)(1 ⫹ i)](1 ⫹ r0 ⫺ ␺)c
⬎ 0.
4b
(81)
This means that the developer’s profit under a presale contract (E(␲ (p*,
p q*
p ))兩presale)
will be larger than that of a spot sale contract (E(␲ (p*,
q*
))
兩
).
Under
this
n
n
spot
circumstance, developers will prefer to use the presale method.
In addition, a buyer’s expected costs will be reduced since:
⌬E(C) ⫽ E(C(p*,
p q*))
p 兩accept ⫺ E(C(p*,
n q*))
n 兩spot
⫽ E(C(p*,
p q*))
p 兩decline ⫺ E(C(p*,
n q*))
n 兩spot
⫽
a ⫹ z ⫹ c␺ (1 ⫹ i) a ⫹ z ⫹ c(1 ⫹ r0)(1 ⫹ i)
⫺
2(1 ⫹ i)
2(1 ⫹ i)
c
2
⫽ ⫺ (1 ⫹ r0 ⫺ ␺) ⬍ 0.
(82)
To see the impact of financing constraints on development scale, developer’s
expected profits and buyer’s expected costs, we obtain:
冋
册
d⌬q d⌬q d␺
c(1 ⫹ i)
⫽
䡠
⫽ ⫺
䡠 [⫺(1 ⫺ d)d] ⬎ 0,
dw
d␺ dw
2b
d⌬E(␲) d⌬E(␲) d␺
⫽
䡠
dw
d␺
dw
⫽
冋
册
c
(a ⫹ z ⫺ c␺ (1 ⫹ i)) 䡠 [⫺(1 ⫺ d)d] ⬎ 0,
2b
⫺
d⌬E(C) d⌬E(C) d␺ c
⫽
䡠
⫽ 䡠 [⫺(1 ⫺ d)d] ⬍ 0.
dw 2
dw
d␺
(83)
(84)
(85)
䡵 Q.E.D.
J R E R
兩
Vo l .
3 0
兩
N o .
3 – 2 0 0 8
3 7 4
兩
C h a n ,
F a n g ,
a n d
Ya n g
Endnotes
1
2
3
4
5
6
7
兩
兩
See Chang and Ward (1993), Wang, Zhou, Chan, and Chau (2000), Hua, Chang, and
Hsieh (2001), Lai, Wang, and Zhou (2004), and Wong, Yiu, Tse, and Chau (2006) for a
discussion of the presale method. The presale method has been used in many cities around
the world, but is particularly popular in several Asian countries/regions, such as Hong
Kong, Taiwan, Korea, Singapore, and Mainland China.
See the August 22, 2006 Los Angeles Times article titled ‘‘State 2nd in Housing Growth,
Census Data Say.’’
This seems especially odd given that Asian researchers and institutions have significantly
increased their influence in top-tier real estate publications, according to an assessment
by Chan, Hardin, Liano, and Yu (2008) of the 1990–2006 real estate literature.
Wang and Zhou (2000), among others, also used a game-theoretical approach to model
the over-building phenomenon in real estate markets.
The option concept has been applied to the study of many issues in real estate markets.
See, for example, Grenadier (1995, 1996), Buetow and Albert (1998), Harrison,
Noordewier, and Ramagopal (2002), Clapham (2003), Lai, Wang, and Zhou (2004),
Svenstrup and Willemann (2006), Wang and Zhou (2006), and Lai, Wang, and Yang
(2007).
The downpayment issues have been examined by Stein (1995) and Leung, Lau, and
Leong (2002). Both papers discuss the downpayment effect in the formation of a positive
relation between property prices and trading volume.
Wang, Young, and Zhou (2002) successfully used a rational model to explain a seemingly
irrational phenomenon (banks foreclosing on properties without negotiations) in the
market. Given this, it might be possible to find a rational model to explain this momentum
buying (or ‘‘last bus’’) phenomenon.
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a n d
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兩
3 7 5
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We acknowledge helpful comments from Rose Lai, David Barker, participants at the
12th AsRES Annual Conference, and the referees of this journal.
Su Han Chan, California State University–Fullerton, Fullerton, CA 92834 or
[email protected].
Fang Fang, Shanghai University of Finance and Economics, Shanghai, P.R. China,
200433 or [email protected].
Jing Yang, California State University–Fullerton, Fullerton, CA 92834 or
[email protected].
J R E R
兩
Vo l .
3 0
兩
N o .
3 – 2 0 0 8